A thicket in a graph $G$ is defined as a set of even circuits such that every edge lies in an even number of them. If $G$ is directed, then each circuit in the thicket has a well defined directed parity. The parity of the thicket is the sum of the parities of its members, and is independent of the orientation of $G$. We study the problem of determining the parity of a thicket $\mathcal{T}$ in terms of structural properties of $\mathcal{T}$. Specifically, we reduce the problem to studying the case where the underlying graph $G$ is cubic. In this case we solve the problem if $|\mathcal{T}| = 3$ or $G$ is bipartite. Some applications to the problem of characterising Pfaffian graphs are also considered.
@article{10_37236_4330,
author = {M. H. de Carvalho and C. H. C. Little},
title = {The parity of a thicket},
journal = {The electronic journal of combinatorics},
year = {2014},
volume = {21},
number = {2},
doi = {10.37236/4330},
zbl = {1300.05248},
url = {http://geodesic.mathdoc.fr/articles/10.37236/4330/}
}
TY - JOUR
AU - M. H. de Carvalho
AU - C. H. C. Little
TI - The parity of a thicket
JO - The electronic journal of combinatorics
PY - 2014
VL - 21
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.37236/4330/
DO - 10.37236/4330
ID - 10_37236_4330
ER -
%0 Journal Article
%A M. H. de Carvalho
%A C. H. C. Little
%T The parity of a thicket
%J The electronic journal of combinatorics
%D 2014
%V 21
%N 2
%U http://geodesic.mathdoc.fr/articles/10.37236/4330/
%R 10.37236/4330
%F 10_37236_4330
M. H. de Carvalho; C. H. C. Little. The parity of a thicket. The electronic journal of combinatorics, Tome 21 (2014) no. 2. doi: 10.37236/4330
